CELLULAR COMPLEXES AS STRUCTURED SEMI-SIMPLICIAL SETS

作     者：HERVE ELTER PASCAL LIENHARDT 

作者机构：I.M.A. — C.N.R. Via De Marini 6 16149 Genova Italy Research fellow — Lavoisier Grant of the French Ministère des Affaires Etrangères. Département d'Informatique S.I.C. (I.R.C.O.M. — U.R.A. C.N.R.S. 356) 40 avenue du Recteur Pineau 86022 Poitiers Cedex France 

出 版 物：《International Journal of Shape Modeling》 

年 卷 期：1994年第1卷第2期

页      面：191-217页

主　　题：Geometric modeling topology-based modeling cellular complexes simplicial sets semi-simplicial sets generalized maps chains of maps 

摘      要：In geometric modeling, numerous combinatorial structures have been proposed for modeling the topology of subdivisions. Often, these structures are deduced from analyses of the topology of cellular subdivisions of E 3 when main combinatorial structures defined in combinatorial topology are simplicial . Here, we present a general framework for the definition of combinatorial structures, for the representation of the topology of triangulable cellular objects. Starting from semi-simplicial sets, which are well known structures in combinatorial topology, we define a simple mechanism, inspired by barycentric triangulation , leading to the definition of a wide class of combinatorial cellular complexes. Other combinatorial structures are deduced for particular subclasses of cellular complexes, using other mechanisms. Our approach is a purely combinatorial one. Advantages and drawbacks are discussed. For instance, relations between these combinatorial structures and geometric objects are deduced from that with semi-simplicial sets. Many important topological properties can be computed on these structures. For particular subclasses of cellular complexes, we can deduce equivalent structures, where less information is explicitly represented, decreasing thus the data storage complexity, and, sometimes, the time complexity of algorithms. One of the main drawbacks is the fact that cells of so-defined cellular complexes are always contractible into a vertex, due to the mechanism inspired from barycentric triangulation. Relations with other works are also discussed.